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A new hydrostatic reconstruction scheme motivated by the wet-dry front

AUTHORS

Guoxian Chen, Sebastian Noelle

ABSTRACT

A key diffculty in the analysis and numerical approximation of the shallow water equations is the non-conservative product of measures due to the gravitational force acting on a sloped bottom. Solutions may be non-unique, and numerical schemes are not only consistent discretizations of the shallow water equations, but they also make a decision how to model the physics. Our derivation is based on infinitesimal singular layers at the cell boundaries, as inspired by [Noelle, Xing, Shu, JCP 2007]. One key step is to separate the singular measures. Another aspect is the reconstruction of the solution variables in the singular layers. We study three reconstructions. The first leads to the well-known scheme of [Audusse, Bristeau, Bouchut, Klein, Perthame, SISC 2004], which introduces the hydrostatic reconstruction. The second is a modification proposed in [Morales, Castro, Pares, AMC 2013], which analyzes if a wave has enough energy to overcome a step. The third is our new scheme, and borrows its structure from the wet-dry front. For a number of cases discussed in recent years, where water runs down a hill, Audusse' scheme converges slowly or fails. Morales' scheme gives a visible improvement. Both schemes are clearly outperformed by our new scheme.

KEYWORDS

shallow water equations, water at rest, well balanced property, wet-dry front, nonconservative products of measures